The continuum parabolic Anderson model with a half-Laplacian and periodic noise

We construct solutions of a renormalized continuum fractional parabolic Anderson model, formally given by ${\mathrm{\partial }}_{t}u=-(-\mathrm{\Delta }{)}^{1\mathrm{/}2}u+\xi u$, where $\xi$ is a periodic spatial white noise. To be precise, we construct limits as $\epsilon \to 0$ to solutions of $\partial_t u_\varepsilon=-(-\Delta)^{1/2}u_\varepsilon+(\xi_\varepsilon-C_\varepsilon)u_\varepsilon$, where ${\xi }_{\epsilon }$ is a mollification of $\xi$ at scale $\epsilon$ and $C_{\epsilon }$ is a logarithmically diverging renormalization constant. We use a simple renormalization scheme based on that of Hairer and Labb'e, “A simple construction of the continuum parabolic Anderson model on ${\mathbf{R}}^2$.”